Topological manifolds and manifold bundles- Lec 06 - Frederic Schuller

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Recommends StefanH · 2016-03 · Mathematics Stack Exchange · on “Two figures, one is not a manifold, but the other is a manifold, but they look both as suffering from the same deficiencies for not being a manifold”

I am looking video lectures [this video] of F. Schuller about space-time geometry, in particular about manifolds. In it, right at the beginning after introducing manifolds he gives a non-example [this video] , i.e. he says that [LINK] is not a manifold as the point where both circles meet (which I made thicker to show it), it is not possible to assign a dimension in concordence with the other points. Intuitively, if I think of this set equipped with the subspace topology derived from the standard topology on $\mathbb R^2$ I am somehow convinced that every open neighbourhood around the problematic point does not resemble an open subset of $\mathbb R$ (despite I do not see how to prove this more formall

Recommends WillG · 2019-01 · Mathematics Stack Exchange · on “Must every manifold bundle be a fiber bundle?”

I have been learning differential geometry from this [LINK] lecture series by Frederic Schuller's. Here [this video] he defines a bundle (of topological manifolds) as A triple, $(E, \pi, M),$ where $E$ and $M$ are topological manifolds and $\pi:E \longrightarrow M$ is a surjective and continuous function. Here [this video] he defines a fiber bundle as A bundle $(E, \pi, M)$ such that $\forall p \in M: \text{preim}_\pi \big(\{p\}\big) \cong F$ for some manifold $F$ . Using these definitions (which I hope are standard), does there exist some bundle of topological manifolds that is not a fiber bundle (i.e., has fibers that are not homeomorphic to each other)? Sidenote, Schuller provides examples here [VI

Cites RagingBull · 2018-03 · Mathematics Stack Exchange · on “implication of a locally trivial bundle”

I am following video lectures by Prof. Fredric Schuller titled "lectures on the geometrical anatomy of theoretical physics".In the following video Manifold bundles [this video] at 1:07:25 he says that locally trivial bundle has an immediate implication that any section can be represented as a map from the base space to the fibre.What does that mean? I have been trying to dissect this statement for a long time but can't get my head around it.Thanks.

Cites Davide Radaelli · 2021-06 · Mathematics Stack Exchange · on “Is this a bundle?”

In Frederic Schuller's lecture series [LINK] Lectures on The Geometrical Anatomy of Theoretical Physics, he gives an example of a bundle [this video] $E\overset{\pi}{\rightarrow}M$ where different points of the base manifold have different fibres: [LINK] Or, as spelled out in Simon Rea's transcription of the lectures, found here [LINK] (p. 39): $$F_p:=\mathrm{preim}_\pi(\{p\}) \cong_\mathrm{top} \begin{cases} S^{1} & p < 0\\ \{ p \} & p = 0\\ [0,1] & p > 0\\ \end{cases}$$ The example is great for intuition. But I don't see how E can be a manifold with such an odd structure (as a reminder, the definition of $E\overset{\pi}{\rightarrow}M$ requires E to be a topological manifold).

Cites diracdeltafunk · 2021-06 · Mathematics Stack Exchange · on “Is this a bundle?”

Here's a link [this video] to that timestamp (working as of June 12, 2021). Since I am not a physicist, I can only imagine (and hope) that these lectures are more helpful than harmful to a physicist's training.